Counting orbits of a product of permutations

A formula is derived for the number of orbits of a product of permutation in terms of the number of orbits of the factors and the nullity of a matrix.     

Turning points in random permutations of two kinds of distinct elements

The mean and variance of the number of turning points in random permutations of two kinds of distinct elements are evaluated. These results are applied to a Wald–Wolfowitz run test.     

On permutations of order dividing a given integer

We give a detailed analysis of the proportion of elements in the symmetric group on n points whose order divides m, for n sufficiently large and m≥n with m=O(n).      

Generation of permutations in lexicographical order

When the permutations are ordered lexicographically there is an ordering number corresponding to each permutation. A relation between the ordering numbers of complementary permutations is shown which can be useful in a computer generation of permutations.     

Half-transitive graphs of order a product of two distinct primes *

A simple undirected graph X is said to be ½-transitive if the automorphism group AutX of X acts transitively onthe vertices and edges, but not on the arcs of X. In this pape we determine all ½-transitive graphs of order a product of two distinct primes     

Tetravalent arc-transitive graphs of order twice a product of two primes

In this article a complete classification of tetravalent arc-transitive graphs of order twice a product of two primes is given.     

Classifying vertex-transitive graphs whose order is a product of two primes

Vertex-transitive graphs whose order is a product of two primes with a primitive automorphism group containing no imprimitive subgroup are classified. Combined with the results of [15] a complete classification of all vertex-transitive graphs whose order is a product of two primes is thus obtained (Theorem 2.1).Supported in part by the Research Council of SloveniaSupported in part by the Italian Ministry of Research (MURST)     

Shuffles of permutations and the Kronecker product

The Kronecker product of two homogeneous symmetric polynomialsP ₁,P ₂ is defined by means of the Frobenius map by the formulaP ₁oP ₂=F(F ⁻¹ P ₁)(F ⁻¹ P ₂). WhenP ₁ andP ₂ are the Schur functionsS _I ,S _J then the resulting productS _I oS _J is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofS _I oS _J with a third Schur functionsS _K gives the so called Kronecker coefficientc _I,J,K =<S _I oS _J ,S _K >. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thec _I,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of <S _I oS _J ,S _K > forS _I a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result. Work supported by NSF grant at the University of California, San Diego.     

On the proportion of permutations of order a multiple of the degree

We study permutations of a set of size n for which the orderis a multiple of n. We prove that, for large n, most such elementslie in one of two families. The first family consists of thosepermutations with a single very large cycle of order dividingn and includes the n-cycles, and the second consists of permutationsfor which the cycles of length dividing n have total lengthsignificantly less than n. This work was inspired by the algorithmicproblem of fast recognition of large symmetric groups actingprimitively on subsets.     

Models of : when two elements are necessarily order automorphic

We are interested in the question of how much the order of a non‐standard model of can determine the model. In particular, for a model M, we want to characterize the complete types of non‐standard elements such that the linear orders and are necessarily isomorphic. It is proved that this set includes the complete types such that if the pair realizes it (in M) then there is an element c such that for all standard n, , , , and . We prove that this is optimal, because if holds, then there is M of cardinality ?₁ for which we get equality. We also deal with how much the order in a model of may determine the addition.     

On the product of two primitive elements of maximal subfields of a finite field

Let denote a finite field with r elements. Let q be a power of a prime, and p₁,p₂, p₃ be distinct primes. Put

     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

On Browkin''''s conjecture about the elements of order five in K_2(Q)

Based on the work of Lenstra, a succinct proof of Browkin's conjecture about the elements of order five in K2(Q) is given.